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Anomalies in time series in Xevents

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DMA definition

Discrete mathematical analysis (DMA) is an approach to studying of multidimensional massifs and time series, based

on modeling of limit in a finite situation, realized in a series of algorithms.

The basis of the finite limit was formed on a more stable character, compared to a mathematic character, of human idea of

discontinuity and stochasticity. Fuzzy mathematics and fuzzy logic are sufficient for modeling of human ideas

and judgments. That was reason why they became technical foundation of DMA.

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Density as limit measure

DMA General Scheme

Fuzzy comparisons ofpositive numbers

Proximity in finite metrical space

Limit in finite metrical space

MultidimensionalDiscrete spaces

Finite time seriesFTS

Recognition of dense subsets:

Crystal. Monolith.

Clusterization:Rodin

Recognition of linear structure:

Tracing

Smooth FTS:Equilibrium

Monotonous FTS

Predicationof FTS:

Forecast

Extremums onFTS

Anomalies on FTS:DRAS, FLARS,

FCARS

ConvexFTS

Fuzzy logic andgeometry on FTS:

Geometry measures12 May 2009

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Classical Set

• Classical set has a clear boundary between elements that do and don’t belong to the set.

• If U is universal set, then a classical subset A U is defined by the membership function μ(x) that takes only two values: μ(x) =1 for the elements belonging to A and μ(x) =0 for elements not belonging to A.

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Fuzzy Set

• A is a fuzzy set in U if there is a map

μ: U 0,1, that shows the degree of inclusion of the element x into the fuzzy set A.

μ(x) is called by membership function of the fuzzy set A.

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Fuzzy Sets Approach in Geophysics

Lofti Zadeh: A human being thinks not in terms of numbers, but rather in terms of fuzzy notions.

Норберт Винер: По-видимому, главное преимущество человека перед компьютером – это его способность оперировать с нечетко очерченными понятиями.

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FLASAR

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La Fournaise volcano

2006, 2005, 2004, 2003, 2002, 2001, 2000, 1998, 1992, 1991, 1990, 1985-88, 1983-84, 1981, 1979, 1977, 1977, 1976, 1975-76, 1973, 1973, 1972, 1966, 1964-65, 1964, 1963, 1961, 1960, 1959, 1958, 1957, 1955-57, 1954, 1953, 1952, 1951, 1950, 1950, 1949, 1948, 1947, 1946, 1945, 1944, 1943, 1942, 1941, 1938-39, 1938, 1937, 1936, 1935, 1933-34, 1932, 1931, 1930, 1929, 1926-27, 1925-26, 1924, 1924, 1921, 1920, 1917, 1915, 1913, 1910, 1909, 1908, 1907, 1905, 1904, 1903, 1902, 1901, 1901, 1900, 1899, 1898, 1898, 1897, 1894, 1890-91, 1889, 1884, 1882, 1878, 1876, 1875, 1874, 1874, 1872, 1871, 1870, 1869, 1868, 1865, 1863-64, 1861, 1860, 1859, 1858-59, 1852, 1851, 1850, 1849, 1848, 1847, 1846, 1845, 1844, 1843, 1842, 1832, 1830, 1824, 1824, 1821, 1820, 1817, 1816, 1815, 1815, 1814, 1813, 1812, 1810, 1809, 1807, 1802, 1801-02, 1800, 1797, 1795, 1794, 1792, 1791, 1789, 1787, 1786, 1784-85, 1776, 1775, 1774, 1772, 1771, 1768, 1766, 1760, 1759, 1753, 1751, 1734, 1734, 1733, 1721, 1709, 1708, 1703, 1672, 1671, 1669, 1649, 1640

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Monitoring of La Fournais volcano, Reunion, France

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DON

300 mV/km

NS

EW

DOS

300 mV/km

NS

EW

MAV

1500 mV/km

NS

EW

27-02-98 01-03-98 03-03-98 05-03-98 07-03-98 09-03-98 11-03-98 13-03-98

Извержение

Recognition SP- anomalyes, connected with volcanic activity

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Eruption

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FCARS: three vision for one time series

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-40

-35

-30

-25

Regression

02-07-0712:00

03-07-0700:00

03-07-0712:00

04-07-0700:00

04-07-0712:00

-40

-35

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Length

-40

-35

-30

-25

Energy

nT

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FCARS: universality

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Agreement “Equilibrium” and “Forecast”

Real time series

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Extremums on time series

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Etna volcano

2005, 2004, 2003, 2002, 2001, 1994, 1993, 1991, 1989, 1988, 1987, 1986, 1985, 1984, 1983, 1981, 1980, 1979, 1978, 1975, 1974, 1971, 1968, 1966, 1959, 1958, 1957, 1955, 1953, 1951, 1950, 1949, 1947, 1946, 1945, 1942, 1940, 1935, 1934, 1931, 1930, 1929, 1928, 1926, 1924, 1923, 1919, 1918, 1917, 1913, 1912, 1911, 1910, 1908, 1899, 1893, 1892, 1891, 1886, 1884, 1883, 1879, 1878, 1874, 1869, 1868, 1865, 1864, 1863, 1857, 1852, 1843, 1842, 1838, 1833, 1832, 1828, 1827, 1822, 1819, 1816, 1811, 1810, 1809, 1803, 1802, 1797, 1792, 1791, 1787, 1781, 1780, 1776, 1770, 1767, 1766, 1764, 1763, 1758, 1755, 1752, 1747, 1744, 1735, 1732, 1723, 1702, 1693, 1689, 1688, 1682, 1669, 1654, 1651, 1646, 1643, 1640, 1634, 1633, 1614, 1610, 1609, 1607, 1603, 1595, 1579, 1578, 1566, 1554, 1550, 1541, 1540, 1537, 1536, 1535, 1533, 1494, 1470, 1447, 1446, 1444, 1408, 1381, 1350, 1334, 1333, 1329, 1321, 1284, 1250, 1222, 1194, 1175, 1169, 1164, 1160, 1157, 1063, 1044, 1004, 0911, 0859, 0814, 0812, 0644, 0604, 0560, 0500, 0417, 0410, 0400, 0252, 0165, 0080, 0072, 0050, 0039, 0010, -0010, -0032, -0036, -0044, -0049, -0056, -0061, -0122, -0126, -0135, -0141, -0350, -0396, -0425, -0479, -0565, -0695, -0735, -1050, -1470, -1500, -2330, -3050, -3390, -3510, -4150, -5150, -619012 May 2009

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Algorithm “Monolith”. Etna volcano

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Interferogram

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Borders

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Algorithm “Monolith”. Etna volcano. Final result

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GIS

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GIS

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Цифровая модель высот (разрешение 30")

Площадная гидрография, Гидрорельеф, Естественные формы рельефа, Изогоны (1:33 000 000)

Фрагмент карты почв (1:1 000 000)

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Literature on DMA to Xevents list

12 May 2009

Gvishiani A.D., Agayan S.M., Bogoutdinov Sh.R., Ledenev A., Zlotnicki J., Bonnin J. Mathematical Methods of Geoinformatics. II. The algorithms of fuzzy logic in the problem of anomalies recognition in time series / / Cybernetics and system analysis. 2003. № 4. p.103-111. Gvishiani A.D., Agayan S.M., Bogoutdinov Sh.R., Zlotnicki J. Algorithms of fuzzy logic in the problem of anomalies recognition in time series / / Sketches of Geophysical Research. By the 75 th anniversary of the Joint Institute of Physics of the Earth RAS. O.Y Schmidt. M.: OIFZ RAS. 2003. p.257-262. Zlotnicki J., Agayan S., Gvishiani A., Bogoutdinov Sh. Telematics and artificial intelligence tools in monitoring of volcanoes // WISTCIS Newsletter. 2003. Vol. 3. November 2002-May 2003. p.58-60. Zlotnicki J., Le Mouel J.-L., Gvishiani A., Agayan S., Mikhailov V., Bogoutdinov Sh., Kanwar R., Yvetot P. Automatic fuzzy-logic recognition of anomalous activity on long geophysical records: Application to electric signals associated with the volcanic activity of La Fournaise volcano (Reunion Island) // Earth and Planetary Science Letters. 2005. Vol. 234. p.261-278.

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Gvishiani A.D., Agayan S.M., Bogoutdinov Sh.R., Tikhotsky S.A., Hinderer J., Bonnin J., Diament M. Algorithm FLARS and recognition of time series anomalies // System Research & Information Technologies. 2004. №. 3. p.7-16. Agayan S.M., Bogoutdinov Sh.R., Gvishiani A.D., Graeva E.M., Zlotnicki J., Rodkin M.V. Investigation of the morphology of the signal based on the algorithms of fuzzy logic / / Geophysical Research. M.: IFZ RAS. 2005. Vol.1. p. 143-155 Zlotnicki J., LeMouel J.-L., Gvishiani A., Agayan S., Mikhailov V., Bogoutdinov Sh. Automatic fuzzy-logic recognition of anomalous activity on long geophysical records. Application to electric signals associated with the volcanic activity of la Fournaise volcano (Réunion Island) // Earth and Planetary Science Letters. 2005. Vol.234. P.261-278. Bogoutdinov Sh.R., Agayan S.M., Gvishiani A.D., Graeva E.M., Rodkin M.V., Zlotnicki J., Le Mouël J.L. Fuzzy logic algorithms in the analysis of electrotelluric data with reference to monitoring of volcanic activity // Izvestiya, Physics of the Solid Earth. MAIK Nauka/Interperiodica distributed exclusively by Springer Science+Business Media LLC . 2007. Vol. 43. p. 597-609.

Literature on DMA to Xevents list

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Gvishiani A.D., Agayan S.M., Bogoutdinov Sh.R. Fuzzy Recognition of Anomalies in Time Series / / Doklady Earth Sciences, June-July 2008, Vol. 421, № 5, p. 838-843. Gvishiani A.D., Agayan S.M., Bogoutdinov Sh.R., Zlotnicki J., Bonnin J. Mathematical Methods of Geoinformatics. III. Fuzzy comparison and recognition of anomalies in time series / / Cybernetics and system analysis. 2008, Vol. 44, № 3, p. 3-18

Literature on DMA to Xevents list